Although the study on the dynamics in the presence of an electromagnetic field is an old issue but due to the following facts it may be still an important issue in the recent technology. The investigation on the ion conducting electrolytic materials is a key area in physics and chemistry [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}. The materials have potential applications in a diverse range of all-solid-state devices, such as rechargeable lithium batteries, flexible electrochromic displays and smart windows [1]}. The properties of the electrolytes are tuned by varying chemical composition to a large extent and hence are adapted to specific needs [6]}, [7]}. High ionic conductivity is needed for optimizing the glassy electrolytes in various applications. Then it would be very interesting if one can tune the ionic conductivity according to specific need by a physical method. In this context, very recent studies [11]}, [12]}, [13]}, [14]}, [15]}, [16]}, [17]}, [18]}, [19]} show that the conductivity of an electrolytic material can be tuned by an applied magnetic field. To tune the conductivity of ions in the solid electrolytes the combination of both magnetic field and time-dependent electric field may be an important choice. Then one may be interested to know the basic dynamics as well as energetics of an analytically solvable model like driven damped harmonic oscillator in the presence of a magnetic field. The driven damped harmonic oscillator is a well studied text book material [20]}, [21]}. At the same time, the dynamics of a particle in the presence of both constant magnetic and electric fields is also a well studied issue in the text book [21]}. This study has been extended in different contexts[23]}, [24]}.
But to the best of our knowledge, the study on the dynamics of periodically driven damped harmonic oscillator in the presence of a magnetic field was not addressed. It does not mean that this issue is not an important one. In other words, the
model study may be a very relevant one in the context of barrier crossing dynamics as mentioned above. At the same time it may also be an important one to explain the refractive
of a dielectric material in the presence of a magnetic field[25]}.
Thus our objective is to explore distinguishable feature (if any) of this dynamics including the energetics and the resonance at the steady state. Then we start with the transients of the two dimensional damped harmonic oscillator (having same frequency along both \(x\)  and \(y\) -directions) in the presence of a magnetic field. \(x(t)\)  and \(y(t)\)  are superposition of two periodic terms in the absence of damping. Thus the motion may be quasi periodic in nature. Then we determine the condition for simple periodic motion. At the same time we determine how the frequencies of the periodic terms in the damped oscillation may depend on the damping strength. These calculations corroborate to the resonance conditions which are determined based on the steady state dynamics for the driven system. Here we find that the magnetic field induces an asymmetric splitting of the spectrum of the output signal with the finite values of the amplitude at the resonance conditions. But the values of the amplitudes at resonance condition become infinite in the absence of damping. It is to be noted here that for this case, the phase shift between the input and output signals at the resonance condition is similar as that of the driven damped harmonic oscillator. It proofs indirectly that the finite value of the amplitude at the resonance condition in the presence of damping is solely due to the dissipation of energy. In other words, the phase shift has no significant role in this context. Major points like these have been included in the conclusion section.
